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# Full text of "Treatise On Analysis Vol-Ii"

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```372       XV    NORMED ALGEBRAS AND SPECTRAL THEORY

follows immediately from the fact that %(x) ^ 0 implies %(x2) ^ 0. The corre-
sponding representation U^ is evidently irreducible.

We denote by H(A) the set of hermitian characters of A; it is a subset of
the product space CA and is closed in the product topology (3.15.1). We give
H(A) the topology induced by the product topology (i.e., the weak topology
(1 2.1 5)). When A is a separable commutative Banach algebra with involution,
having a unit element e 7^ 0, the space H(A) is a compact metrizable subspace
of X(A) (15.3.2). In this situation it can happen that H(A) =£ X(A) (Section
15.4, Problem 3), but H(A) is equal to X(A) if A is a separable star algebra
(15.4.14).

We shall show that the bitraces g satisfying the conditions at the begin-
ning of this section can all be obtained, by a canonical process of " integra-
tion", from hermitian characters:

(1 5.9.2) (Plancherel-Godement theorem) Let g be a bitrace on a com-
mutative algebra A with involution, satisfying (U) and (N), and such that the
prehilbert space A/ng and the star algebra &\$ g a Jzf (H^) are separable.

(I) We can define canonically: (1) a subspace S^ <9/H(A), whose closure in
CA fs either Sg or Sgu {0} and is metrizable and compact (so that S^ is locally
compact, metrizable and separable) ; (2) a positive measure mg on Sg , with the
following properties :

(i)   For each x e A, the function %*-+x(i) = %(x) belongs to &c(Sg, mg)
and we have

mg(x)

(15.9.2.1)        g(x, y) = f fay*) dmg(X) - f
Js*                           Js*

for all x, y in A.

(ii) As x runs through A, the set of functions x is contained in #c(Sfl)
(13.20.6) and is dense in this Banach space.

(iii)   The support of the measure mg is the whole ofSg .

(iv) The mapping x\-+x factor izes into x\-~*ng(x)&x, and the mapping
T0 ofA/ng into #2(8^) extends to an isomorphism T of the Hilbert space Hg onto
L|(S, , mg) such that for allxeA we have Ug(x) = T~lM(x)T, where M(x) is
multiplication by the class ofx in L%(\$g,mg) (cf. (13.21.5)). Also

for all 1 e C and x e A.

(II)   Conversely, let She a subspace of H(A) such that S u {0} is compact
and metrizable, and let m be a positive measure on S, with support equal to S,
```